“A program is a collection of projects, where the value is in the overall deliverable. Yes, each project may have a deliverable that’s valuable, but the value to the organization is when all the sub-projects get together and deliver their product.”

That is certainly one definition, but there are others as well. A program may contain several related projects, whereby managing them collectively provides benefits to the organization. An example might be the building of several houses in a development. Each house is a single product, but by coordinating purchasing and construction across many products within the program, substantial efficiencies are realized. Another example is a program that manages a single product throughout the product lifecycle, such as software over multiple releases or a model of aircraft. There may be many projects in the program, but each project delivers a valuable product or enhances an already existing product.

One question about terminology. If a program is made up of a collection of projects, why use the term sub-project? This would imply, although you don’t state it, that the projects that make up the program must be further decomposed so that projects are in turn made up of sub-projects. Maybe that was your intention.